| L(s) = 1 | + (−0.951 + 0.309i)2-s + (0.821 + 1.52i)3-s + (0.809 − 0.587i)4-s + (0.866 − 0.5i)5-s + (−1.25 − 1.19i)6-s + (0.122 + 1.16i)7-s + (−0.587 + 0.809i)8-s + (−1.64 + 2.50i)9-s + (−0.669 + 0.743i)10-s + (0.519 + 0.231i)11-s + (1.56 + 0.750i)12-s + (0.447 + 2.10i)13-s + (−0.476 − 1.07i)14-s + (1.47 + 0.909i)15-s + (0.309 − 0.951i)16-s + (−5.00 + 2.22i)17-s + ⋯ |
| L(s) = 1 | + (−0.672 + 0.218i)2-s + (0.474 + 0.880i)3-s + (0.404 − 0.293i)4-s + (0.387 − 0.223i)5-s + (−0.511 − 0.488i)6-s + (0.0462 + 0.440i)7-s + (−0.207 + 0.286i)8-s + (−0.549 + 0.835i)9-s + (−0.211 + 0.235i)10-s + (0.156 + 0.0697i)11-s + (0.450 + 0.216i)12-s + (0.124 + 0.583i)13-s + (−0.127 − 0.285i)14-s + (0.380 + 0.234i)15-s + (0.0772 − 0.237i)16-s + (−1.21 + 0.540i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.644 - 0.764i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.644 - 0.764i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.513426 + 1.10420i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.513426 + 1.10420i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.951 - 0.309i)T \) |
| 3 | \( 1 + (-0.821 - 1.52i)T \) |
| 5 | \( 1 + (-0.866 + 0.5i)T \) |
| 31 | \( 1 + (-4.19 - 3.65i)T \) |
| good | 7 | \( 1 + (-0.122 - 1.16i)T + (-6.84 + 1.45i)T^{2} \) |
| 11 | \( 1 + (-0.519 - 0.231i)T + (7.36 + 8.17i)T^{2} \) |
| 13 | \( 1 + (-0.447 - 2.10i)T + (-11.8 + 5.28i)T^{2} \) |
| 17 | \( 1 + (5.00 - 2.22i)T + (11.3 - 12.6i)T^{2} \) |
| 19 | \( 1 + (-3.58 - 0.762i)T + (17.3 + 7.72i)T^{2} \) |
| 23 | \( 1 + (5.52 + 4.01i)T + (7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (-3.19 - 9.82i)T + (-23.4 + 17.0i)T^{2} \) |
| 37 | \( 1 + (1.33 + 0.771i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-1.22 - 1.10i)T + (4.28 + 40.7i)T^{2} \) |
| 43 | \( 1 + (1.07 - 5.07i)T + (-39.2 - 17.4i)T^{2} \) |
| 47 | \( 1 + (-0.408 - 0.132i)T + (38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (-0.499 + 4.75i)T + (-51.8 - 11.0i)T^{2} \) |
| 59 | \( 1 + (2.81 - 2.53i)T + (6.16 - 58.6i)T^{2} \) |
| 61 | \( 1 + 5.66iT - 61T^{2} \) |
| 67 | \( 1 + (-6.10 - 10.5i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (10.1 + 1.06i)T + (69.4 + 14.7i)T^{2} \) |
| 73 | \( 1 + (-1.02 + 2.29i)T + (-48.8 - 54.2i)T^{2} \) |
| 79 | \( 1 + (-0.346 - 0.777i)T + (-52.8 + 58.7i)T^{2} \) |
| 83 | \( 1 + (2.23 - 2.47i)T + (-8.67 - 82.5i)T^{2} \) |
| 89 | \( 1 + (10.6 - 7.70i)T + (27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (-0.672 + 0.488i)T + (29.9 - 92.2i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.18488614902817741175219828452, −9.453331262982151281326745357044, −8.678351624776853677726060319020, −8.339773405956618814695337908692, −7.01455394166460356436973842916, −6.11762011400433583746129778616, −5.10438267932468304820287835054, −4.19446807755925260876459249140, −2.85673286535781818394934884947, −1.75634827344580054230572768103,
0.66152020657638502240078811866, 2.02569113148743024836264448440, 2.92238468975489052054360413494, 4.13168462574402786700689195917, 5.76205191383401318120930590689, 6.53992265699067173526625838801, 7.43969939808970150134339440605, 7.998202131754329928045693142654, 8.946187116749453345914973383605, 9.679912211988513059821204640971
